#488590

order	512
nilpotency class	4
isoclinism	isoclinic class of #488539
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#60915	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#424 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

687 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 80
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 80 points:

(1,2)(3,6)(4,7)(5,8)(9,15)(10,13)(11,14)(12,16)(17,18,24,30)(19,31,35,60)(20,36,40,57)(21,27,23,29)(22,42,45,55)(25,47,51,34)(26,37,53,39)(28,41,56,44)(32,61,65,77)(33,59,46,54)(38,69,72,52)(43,49,73,50)(48,70,76,80)(58,62,79,64)(63,66,74,75)(67,68,78,71);(1,3)(2,6)(4,9)(5,10)(7,12)(8,13)(11,16)(14,15)(17,19)(18,25)(20,32)(21,33)(22,43)(23,46)(24,35)(26,48)(27,49)(28,54)(29,50)(30,51)(31,42)(34,44)(36,66)(37,62)(38,70)(39,64)(40,65)(41,47)(45,73)(52,77)(53,76)(55,60)(56,59)(57,75)(58,67)(61,69)(63,71)(68,74)(72,80)(78,79);(1,4,5,11)(2,7,8,14)(3,9,10,16)(6,12,13,15)(17,20)(18,26)(19,32)(21,37)(22,38)(23,39)(24,40)(25,48)(27,36)(28,52)(29,57)(30,53)(31,58)(33,62)(34,63)(35,65)(41,68)(42,67)(43,70)(44,71)(45,72)(46,64)(47,74)(49,66)(50,75)(51,76)(54,77)(55,78)(56,69)(59,61)(60,79)(73,80);(2,8)(6,13)(7,14)(12,15)(17,21,24,23)(18,27,30,29)(19,33,35,46)(20,37,40,39)(22,41,45,44)(25,49,51,50)(26,36,53,57)(28,42,56,55)(31,59,60,54)(32,62,65,64)(34,43,47,73)(38,68,72,71)(48,66,76,75)(52,67,69,78)(58,61,79,77)(63,70,74,80)

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=i,e^2=gi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fi,g^a=gi,e^b=eg,f^b=fh,g^b=gi,e^c=eh,f^c=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 395786);